
SHGEQZ(l) ) SHGEQZ(l)
NAME
SHGEQZ  implement a single/doubleshift version of the QZ method for finding the gener
alized eigenvalues w(j)=(ALPHAR(j) + i*ALPHAI(j))/BETAR(j) of the equation det( A  w(i)
B ) = 0 In addition, the pair A,B may be reduced to generalized Schur form
SYNOPSIS
SUBROUTINE SHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA,
Q, LDQ, Z, LDZ, WORK, LWORK, INFO )
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), Q( LDQ,
* ), WORK( * ), Z( LDZ, * )
PURPOSE
SHGEQZ implements a single/doubleshift version of the QZ method for finding the general
ized eigenvalues w(j)=(ALPHAR(j) + i*ALPHAI(j))/BETAR(j) of the equation det( A  w(i) B )
= 0 In addition, the pair A,B may be reduced to generalized Schur form: B is upper trian
gular, and A is block upper triangular, where the diagonal blocks are either 1by1 or
2by2, the 2by2 blocks having complex generalized eigenvalues (see the description of
the argument JOB.)
If JOB='S', then the pair (A,B) is simultaneously reduced to Schur form by applying one
orthogonal tranformation (usually called Q) on the left and another (usually called Z) on
the right. The 2by2 uppertriangular diagonal blocks of B corresponding to 2by2
blocks of A will be reduced to positive diagonal matrices. (I.e., if A(j+1,j) is non
zero, then B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1) will be positive.)
If JOB='E', then at each iteration, the same transformations are computed, but they are
only applied to those parts of A and B which are needed to compute ALPHAR, ALPHAI, and
BETAR.
If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the orthogonal transformations used to
reduce (A,B) are accumulated into the arrays Q and Z s.t.:
Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)*
Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)*
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241256.
ARGUMENTS
JOB (input) CHARACTER*1
= 'E': compute only ALPHAR, ALPHAI, and BETA. A and B will not necessarily be put
into generalized Schur form. = 'S': put A and B into generalized Schur form, as
well as computing ALPHAR, ALPHAI, and BETA.
COMPQ (input) CHARACTER*1
= 'N': do not modify Q.
= 'V': multiply the array Q on the right by the transpose of the orthogonal tran
formation that is applied to the left side of A and B to reduce them to Schur
form. = 'I': like COMPQ='V', except that Q will be initialized to the identity
first.
COMPZ (input) CHARACTER*1
= 'N': do not modify Z.
= 'V': multiply the array Z on the right by the orthogonal tranformation that is
applied to the right side of A and B to reduce them to Schur form. = 'I': like
COMPZ='V', except that Z will be initialized to the identity first.
N (input) INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER It is assumed that A is already upper triangular in rows
and columns 1:ILO1 and IHI+1:N. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0,
if N=0.
A (input/output) REAL array, dimension (LDA, N)
On entry, the NbyN upper Hessenberg matrix A. Elements below the subdiagonal
must be zero. If JOB='S', then on exit A and B will have been simultaneously
reduced to generalized Schur form. If JOB='E', then on exit A will have been
destroyed. The diagonal blocks will be correct, but the offdiagonal portion will
be meaningless.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B (input/output) REAL array, dimension (LDB, N)
On entry, the NbyN upper triangular matrix B. Elements below the diagonal must
be zero. 2by2 blocks in B corresponding to 2by2 blocks in A will be reduced
to positive diagonal form. (I.e., if A(j+1,j) is nonzero, then
B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1) will be positive.) If JOB='S', then
on exit A and B will have been simultaneously reduced to Schur form. If JOB='E',
then on exit B will have been destroyed. Elements corresponding to diagonal
blocks of A will be correct, but the offdiagonal portion will be meaningless.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
ALPHAR (output) REAL array, dimension (N)
ALPHAR(1:N) will be set to real parts of the diagonal elements of A that would
result from reducing A and B to Schur form and then further reducing them both to
triangular form using unitary transformations s.t. the diagonal of B was nonnega
tive real. Thus, if A(j,j) is in a 1by1 block (i.e., A(j+1,j)=A(j,j+1)=0), then
ALPHAR(j)=A(j,j). Note that the (real or complex) values (ALPHAR(j) +
i*ALPHAI(j))/BETA(j), j=1,...,N, are the generalized eigenvalues of the matrix
pencil A  wB.
ALPHAI (output) REAL array, dimension (N)
ALPHAI(1:N) will be set to imaginary parts of the diagonal elements of A that
would result from reducing A and B to Schur form and then further reducing them
both to triangular form using unitary transformations s.t. the diagonal of B was
nonnegative real. Thus, if A(j,j) is in a 1by1 block (i.e.,
A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=0. Note that the (real or complex) values
(ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the generalized eigenvalues of
the matrix pencil A  wB.
BETA (output) REAL array, dimension (N)
BETA(1:N) will be set to the (real) diagonal elements of B that would result from
reducing A and B to Schur form and then further reducing them both to triangular
form using unitary transformations s.t. the diagonal of B was nonnegative real.
Thus, if A(j,j) is in a 1by1 block (i.e., A(j+1,j)=A(j,j+1)=0), then
BETA(j)=B(j,j). Note that the (real or complex) values (ALPHAR(j) +
i*ALPHAI(j))/BETA(j), j=1,...,N, are the generalized eigenvalues of the matrix
pencil A  wB. (Note that BETA(1:N) will always be nonnegative, and no BETAI is
necessary.)
Q (input/output) REAL array, dimension (LDQ, N)
If COMPQ='N', then Q will not be referenced. If COMPQ='V' or 'I', then the trans
pose of the orthogonal transformations which are applied to A and B on the left
will be applied to the array Q on the right.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or 'I', then LDQ >=
N.
Z (input/output) REAL array, dimension (LDZ, N)
If COMPZ='N', then Z will not be referenced. If COMPZ='V' or 'I', then the
orthogonal transformations which are applied to A and B on the right will be
applied to the array Z on the right.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or 'I', then LDZ >=
N.
WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If LWORK = 1, then a workspace query is assumed; the routine only calculates the
optimal size of the WORK array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (A,B) is not in Schur form, but
ALPHAR(i), ALPHAI(i), and BETA(i), i=INFO+1,...,N should be correct. =
N+1,...,2*N: the shift calculation failed. (A,B) is not in Schur form, but
ALPHAR(i), ALPHAI(i), and BETA(i), i=INFON+1,...,N should be correct. > 2*N:
various "impossible" errors.
FURTHER DETAILS
Iteration counters:
JITER  counts iterations.
IITER  counts iterations run since ILAST was last
changed. This is therefore reset only when a 1by1 or
2by2 block deflates off the bottom.
LAPACK version 3.0 15 June 2000 SHGEQZ(l) 
